If by surface you mean a 2-dimensional manifold, then no. The Nash embedding threorms give upper bounds for how big the embedding target dimension needs to be f

If by surface you mean a 2-dimensional manifold, then no. The Nash embedding threorms give upper bounds for how big the embedding target dimension needs to be for any n-dimensional manifold. In the case of 2-dimensional surfaces, at worst you'll need 51 = 2×(2+1)×(3×2+11)/2 dimensions. Mind this is for isometric embeddings (ones that don't squish things around in a sense), which is kind of the most stringent constraint possible. If you allow squishing things around, then the upper bounds get even smaller. https://en.wikipedia.org/wiki/Nash_embedding_theorems#Ck_embedding_theorem https://en.wikipedia.org/wiki/Nash_embedding_theorems#Ck_emb...